Domain decomposition and fictitious domain methods with distributed Lagrange multipliers

and a piece-wise constant diffusion coefficient . The above restrictions are imposed for the sake of simplicity. The generalizations of the algorithms and theoretical results to more complicated equations, domains, and boundary conditions are obvious. Let be a triangular/tetrahedral partitioning of , and ! be the corresponding piecewise linear finite element subspace of "$# % . We shall always assume in this paper that is a shape-regular mesh. Then the classical finite element method ! & ' ( ) *+ () , )! (3)